sampled-data piecewise affine slab systems: a time-delay approach
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TRANSCRIPT
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Sampled-Data Piecewise Affine Slab Systems:
A Time-Delay Approach
Behzad Samadi Luis Rodrigues
Department of Mechanical and Industrial Engineering
Concordia University
ACC 2008, Seattle, WA
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Outline of Topics
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Practical Motivation
c©Quanser
Memoryless Nonlinearities
Saturation Dead Zone Coulomb &Viscous Friction
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Motivational example
Toycopter, a 2 DOF helicopter model
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Motivational example
Pitch model of the experimental helicopter:
x1 =x2
x2 =1
Iyy(−mheli lcgxg cos(x1)−mheli lcgzg sin(x1)− FkM sgn(x2)
− FvMx2 + u)
where x1 is the pitch angle and x2 is the pitch rate.
Nonlinear part:
f (x1) = −mheli lcgxg cos(x1)−mheli lcgzg sin(x1)
PWA part:f (x2) = −FkM sgn(x2)
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Sampled-Data PWA Systems: A Time-Delay Approach
x1
f(x
1)
f (x1)
f (x1)
-3.1416 -1.885 -0.6283 0.6283 1.885 3.1416-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
PWA approximation - Helicopter model
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Objective
To propose a stability analysis method for sampled-data PWAsystems using
convex optimization
time-delay approach
Continuous−time
PWA systems
PWA controller
Hold
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Piecewise Affine Systems
PWA systems are in general nonsmooth nonlinear systems.
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Piecewise Affine Systems
PWA systems are in general nonsmooth nonlinear systems.
Controller synthesis methods for PWA systems
Hassibi and Boyd (1998) - Quadratic stabilization and controlof piecewise linear systems - Limited to piecewise linearcontrollers for PWA systems with one variable in the domain ofnonlinearityJohansson and Rantzer (2000) - Piecewise linear quadraticoptimal control - No guarantee for stabilityFeng (2002) - Controller design and analysis of uncertainpiecewise linear systems - All local subsystems should be stableRodrigues and How (2003) - Observer-based control ofpiecewise affine systems - Bilinear matrix inequality
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Sampled-Data PWA Systems: A Time-Delay Approach
PWA slab system
x = Aix + ai + Bu, for x ∈ Ri
with the region Ri defined as
Ri = x | σi < CRx < σi+1,
where CR ∈ R1×n and σi for i = 1, . . . ,M + 1 are scalars such
thatσ1 < σ2 < . . . < σM+1
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Sampled-Data PWA Systems: A Time-Delay Approach
PWA slab system
x = Aix + ai + Bu, for x ∈ Ri
with the region Ri defined as
Ri = x | σi < CRx < σi+1,
where CR ∈ R1×n and σi for i = 1, . . . ,M + 1 are scalars such
thatσ1 < σ2 < . . . < σM+1
Continuous-time PWA controller
u(t) = Kix(t) + ki , x(t) ∈ Ri
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Sampled-Data PWA Systems: A Time-Delay Approach
Lyapunov-Krasovskii functional:
V (xs , ρ) := V1(x) + V2(xs , ρ) + V3(xs , ρ)
where
xs(t) :=
[
x(t)x(tk)
]
, tk ≤ t < tk+1
V1(x) := xTPx
V2(xs , ρ) :=
∫ 0
−τM
∫ t
t+r
xT(s)Rx(s)dsdr
V3(xs , ρ) := (τM − ρ)(x(t)− x(tk))TX (x(t)− x(tk))
and P , R and X are positive definite matrices.
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Sampled-Data PWA Systems: A Time-Delay Approach
The closed-loop system can be rewritten as
x(t) = Aix(t) + ai + B(Kix(tk) + ki ) + Bw ,
for x(t) ∈ Ri and x(tk) ∈ Rj where
w(t) = (Kj − Ki )x(tk) + (kj − ki ), x(t) ∈ Ri , x(tk) ∈ Rj
The input w(t) is a result of the fact that x(t) and x(tk) arenot necessarily in the same region.
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Sampled-Data PWA Systems: A Time-Delay Approach
Theorem (1)
For the sampled-data PWA system, assume there exist symmetric
positive matrices P ,R ,X and matrices Ni for i = 1, . . . ,M such
that the conditions are satisfied and let there be constants ∆K and
∆k such that
‖w‖ ≤ ∆K‖x(tk)‖+∆k
Then, all the trajectories of the sampled-data PWA system in Xconverge to the following invariant set
Ω = xs | V (xs , ρ) ≤ σaµ2θ + σb
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Sampled-Data PWA Systems: A Time-Delay Approach
for all i ∈ I(0),
Ωi + τMM1i + τMM2i < 0
Ωi + τMM1i τM
[
Ni
0
]
τM[
NTi 0
]
−τMR
< 0
for all i /∈ I(0), Λi ≻ 0,
Ωi + τMM1i + τMM2i < 0
Ωi + τMM1i τM
Ni
00
τM[
NTi 0 0
]
−τMR
< 0
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Sampled-Data PWA Systems: A Time-Delay Approach
Solving an optimization problem to maximize τM subject to theconstraints of the main theorem and η > γ > 1 leads to
τ⋆M = 0.2193
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Sampled-Data PWA Systems: A Time-Delay Approach
x1
x2
-3 -2 -1 0 1 2 3-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Sampled data PWA controller for Ts = 0.2193
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Sampled-Data PWA Systems: A Time-Delay Approach
x1
x2
-3 -2 -1 0 1 2 3-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Continuous time PWA controller
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Summary of the contributions:
Formulating stability analysis of sampled-data PWA slabsystems as a convex optimization problem
Future work:
Formulating controller synthesis for sampled-data PWA slabsystems as a convex optimization problem